Is an Anti-Symmetric Relation also Reflexive? According to the definition of an Anti-Symmetric Relation if xRy and yRx then x = y
Which means, effectively, x is in relation with itself. Does this mean that anti-symmetry implies reflexive property as well?
 A: Your definition is wrong. The relation $R$ is antisymmetric if, whenever $x\mathrel{R}y$ and $y\mathrel R x$ it holds that $x=y$.
An example of a relation that is antisymmetric but not reflexive is $>$ on the set of integers.
A: Not really. For example the empty relation is anti-symmetric, but is not reflexive unless the underlying set is empty as well.
I hope this helps $\ddot\smile$.
A: A relation that is antisymmetric but not reflexive is said to be "strongly antisymmetric" or "asymmetric". 
This implies :
$$(xRy) \implies (\neg(yRx))$$
As if $(xRy)$ and $(yRx)$, then $x=y$ but $x\not R x$ because $R$ is not reflexive (which mean you actually can't apply antisymetry to deduce equality).
$>$ and $<$ are the most common examples.
A: Anti-symmetry states that if xRy and yRx, then x=y.
Now, concerning a relation like >, it is clearly impossible to meet the conditions xRy and yRx (see why? no number can be strictly greater and strictly smaller than an other number at the same time).
Whenever, the conditions of some property cannot be met then we consider that it's true and that the property holds (because "A implies B" is always true if A is false). So > is anti-symmetric. Hope that helps !
